Difference between revisions of "2019 AMC 10B Problems"
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==Problem 22== | ==Problem 22== | ||
− | Raashan, Sylvia, and Ted play the following game. Each starts with <math> \$1</math>. A bell rings every <math>15</math> seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives <math>\$1</math> to that player. What is the probability that after the bell has rung <math>2019</math> times, each player will have <math>\$1</math>? (For example, Raashan and Ted may each decide to give <math>\$1</math> to Sylvia, and Sylvia may decide to give | + | Raashan, Sylvia, and Ted play the following game. Each starts with <math> \$1</math>. A bell rings every <math>15</math> seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives <math>\$1</math> to that player. What is the probability that after the bell has rung <math>2019</math> times, each player will have <math>\$1</math>? (For example, Raashan and Ted may each decide to give <math>\$1</math> to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have <math>\$0</math>, Sylvia will have <math>\$2</math>, and Ted will have <math>\$1</math>, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their <math> \$1</math> to, and the holdings will be the same at the end of the second round.) |
<math>\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}</math> | <math>\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}</math> |
Revision as of 19:31, 30 December 2020
2019 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Alicia had two containers. The first was full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was
full of water. What is the ratio of the volume of the first container to the volume of the second container?
Problem 2
Consider the statement, "If is not prime, then
is prime." Which of the following values of
is a counterexample to this statement?
Problem 3
In a high school with students,
of the seniors play a musical instrument, while
of the non-seniors do not play a musical instrument. In all,
of the students do not play a musical instrument. How many non-seniors play a musical instrument?
Problem 4
All lines with equation such that
form an arithmetic progression pass through a common point. What are the coordinates of that point?
Problem 5
Triangle lies in the first quadrant. Points
,
, and
are reflected across the line
to points
,
, and
, respectively. Assume that none of the vertices of the triangle lie on the line
. Which of the following statements is not always true?
Triangle
lies in the first quadrant.
Triangles
and
have the same area.
The slope of line
is
.
The slopes of lines
and
are the same.
Lines
and
are perpendicular to each other.
Problem 6
There is a real such that
. What is the sum of the digits of
?
Problem 7
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either pieces of red candy,
pieces of green candy,
pieces of blue candy, or
pieces of purple candy. A piece of purple candy costs
cents. What is the smallest possible value of
?
Problem 8
The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?
Problem 9
The function is defined by
for all real numbers
, where
denotes the greatest integer less than or equal to the real number
. What is the range of
?
Problem 10
In a given plane, points and
are
units apart. How many points
are there in the plane such that the perimeter of
is
units and the area of
is
square units?
Problem 11
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar the ratio of blue to green marbles is
, and the ratio of blue to green marbles in Jar
is
. There are
green marbles in all. How many more blue marbles are in Jar
than in Jar
?
Problem 12
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than ?
Problem 13
What is the sum of all real numbers for which the median of the numbers
and
is equal to the mean of those five numbers?
Problem 14
The base-ten representation for is
, where
,
, and
denote digits that are not given. What is
?
Problem 15
Right triangles and
have areas 1 and 2, respectively. A side of
is congruent to a side of
, and a different side of
is congruent to a different side of
. What is the square of the product of the other (third) sides of
and
?
Problem 16
In with a right angle at
point
lies in the interior of
and point
lies in the interior of
so that
and the ratio
What is the ratio
Problem 17
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin is
for
What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
Problem 18
Henry decides one morning to do a workout, and he walks of the way from his home to his gym. The gym is
kilometers away from Henry's home. At that point, he changes his mind and walks
of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks
of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked
of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point
kilometers from home and a point
kilometers from home. What is
?
Problem 19
Let be the set of all positive integer divisors of
How many numbers are the product of two distinct elements of
Problem 20
As shown in the figure, line segment is trisected by points
and
so that
Three semicircles of radius
and
have their diameters on
and are tangent to line
at
and
respectively. A circle of radius
has its center on
The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
where
and
are positive integers and
and
are relatively prime. What is
?
Problem 21
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?
Problem 22
Raashan, Sylvia, and Ted play the following game. Each starts with . A bell rings every
seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives
to that player. What is the probability that after the bell has rung
times, each player will have
? (For example, Raashan and Ted may each decide to give
to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have
, Sylvia will have
, and Ted will have
, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their
to, and the holdings will be the same at the end of the second round.)
Problem 23
Points and
lie on circle
in the plane. Suppose that the tangent lines to
at
and
intersect at a point on the
-axis. What is the area of
?
Problem 24
Define a sequence recursively by and
for all nonnegative integers
Let
be the least positive integer such that
In which of the following intervals does
lie?
Problem 25
How many sequences of s and
s of length
are there that begin with a
, end with a
, contain no two consecutive
s, and contain no three consecutive
s?
See also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2019 AMC 10A Problems |
Followed by 2020 AMC 10A Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.